Chapter 1

You inherit \(\$ 18,750\) with the stipulation that for the first year the money must be placed in two investments paying \(10 \%\) and \(12 \%\) annual interest, respectively. How much should be invested at each rate if the total interest earned for the year is to be \(\$ 2117 ?\)

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$\frac{x}{4}-\frac{3}{5} \leq \frac{x}{2}+1$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{8 x}{x+1}=4-\frac{8}{x+1} $$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$\sqrt{-12}(\sqrt{-4}-\sqrt{2})$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{1}{x-3}<1 $$

Solve each equation in by making an appropriate substitution. $$ 4 x^{4}=13 x^{2}-9 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+4 x=12 $$

Things did not go quite as planned. You invested \(\$ 12,000\), part of it in stock that paid \(14 \%\) annual interest. However, the rest of the money suffered a \(6 \%\) loss. If the total annual income from both investments was \(\$ 680,\) how much was invested at each rate?

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$\frac{3 x}{10}+1 \geq \frac{1}{5}-\frac{x}{10}$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{2}{x-2}=\frac{x}{x-2}-2 $$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(3 \sqrt{-5})(-4 \sqrt{-12})$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x-2}{x+2} \leq 2 $$

Solve each equation in by making an appropriate substitution. $$ x-13 \sqrt{x}+40=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-6 x-11=0 $$

The length of the rectangular tennis court at Wimbledon is 6 feet longer than twice the width. If the court's perimeter is 228 feet, what are the court's dimensions?

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$1-\frac{x}{2}>4$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{3}{2 x-2}+\frac{1}{2}=\frac{2}{x-1} $$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(3 \sqrt{-7})(2 \sqrt{-8})$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x}{x+2} \geq 2 $$

Solve each equation in by making an appropriate substitution. $$ 2 x-7 \sqrt{x}-30=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-2 x-5=0 $$

A rectangular soccer field is twice as long as it is wide. If the perimeter of the soccer field is 300 yards, what are its dimensions?

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{3}{x+3}=\frac{5}{2 x+6}+\frac{1}{x-2} $$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{3}{x+3}>\frac{3}{x-2} $$

Solve each equation in by making an appropriate substitution. $$ x^{-2}-x^{-1}-20=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+4 x+1=0 $$

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$\frac{x-4}{6} \geq \frac{x-2}{9}+\frac{5}{18}$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{3}{x+2}+\frac{2}{x-2}=\frac{8}{(x+2)(x-2)} $$

Explain how to add complex numbers. Provide an example with your explanation.

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+6 x-5=0 $$

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$\frac{4 x-3}{6}+2 \geq \frac{2 x-1}{12}$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{5}{x+2}+\frac{3}{x-2}=\frac{12}{(x+2)(x-2)} $$

Explain how to multiply complex numbers and give an example.

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x^{2}-x-2}{x^{2}-4 x+3}>0 $$

Solve each equation in by making an appropriate substitution. $$ x^{2 / 3}-x^{1 / 3}-6=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+3 x-1=0 $$

An automobile repair shop charged a customer \(\$ 448\) listing \(\$ 63\) for parts and the remainder for labor. If the cost of labor is \(\$ 35\) per hour, how many hours of labor did it take to repair the car?

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$4(3 x-2)-3 x<3(1+3 x)-7$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{2}{x+1}-\frac{1}{x-1}=\frac{2 x}{x^{2}-1} $$

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x^{2}-3 x+2}{x^{2}-2 x-3}>0 $$

Solve each equation in by making an appropriate substitution. $$ 2 x^{2 / 3}+7 x^{1 / 3}-15=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-3 x-5=0 $$

A repair bill on a yacht came to \(\$ 1603,\) including \(\$ 532\) for parts and the remainder for labor. If the cost of labor is \(\$ 63\) per hour, how many hours of labor did it take to repair the yacht?

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$3(x-8)-2(10-x)>5(x-1)$$

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{4}{x+5}+\frac{2}{x-5}=\frac{32}{x^{2}-25} $$

Explain how to divide complex numbers. Provide an example with your explanation.

Use the position formula $$ s=-16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{0}=\text { initial velocity, } s_{0}=\text { initial position, } t=\text { time }\right)\) to answer Exercises \(49-52 .\) If necessary, round answers to the nearest hundredth of a second. A projectile is fired straight upward from ground level with an initial velocity of 80 feet per second. During which interval of time will the projectile's height exceed 96 feet?

What is the rectangular coordinate system?